Key slides on production, supply, etc.

Key elements to focus on in lecture (handout covered on your own)

  1. Production function: returns to scale; how to compute it, marginal product of each input, how to compute it

  2. Cost-minimisation problem, cost-minimising input choice: for interior solution isocost and isoquant are parallel; MRTS = input price ratio (see ‘conditional input demand’)

  3. The ‘cost function’ and the total cost curve… with IRS, DRS… (Total costs, marginal costs, average costs, fixed costs: on your own)

  4. Firm’s profit-maximisation: An unconstrained problem… optimal q where MR(q)=mc(q) if TR>TC; inverse-elasticity rule; Shut-down decision

(On your own: what is ‘Economic profit?’)

  1. Perfect competition: Definition; Entry and exit of firms yield zero profit in the LR; ‘Price-taker’ firms (what leads to this, what it means); firm’s supply curve under perfect competition; Market supply curve under PC, short and LR; Welfare consequences of perfect competition (CS)

  2. Also: reminder about estimating a supply curve

The Firm’s Production function

The Firm’s Production function

\[q = f( K, L, M, ...)\]

\[q = f(K, L)\]

e.g., ‘Cobb douglas with particular proportions and constant returns to scale’

\[q =2L^{1/3}K^{2/3}\]


or ‘Leonteif’

\[y=min\{a_1 x_1, a_2 x_2, ...a_n x_n \}\] . . .

Or ‘two-input Constant Elasticity of Substitution’:

\[x_1^\rho + x_2^\rho)^(1/\rho)\]


Marginal product of each input

Additional output from adding +1 unit of an input, holding other inputs constant.

Marginal Product of Labour—\(MP_L\): Slope of production function in units of labour (holding capital etc constant)

\[MP_L=\frac{\partial}{\partial L}f(K,L)\]


E.g., for \(q = f(K,L) = 2L^{1/3}K^{2/3}\)\(MP_L(K,L) =\) ?

 Check, does this production function have ‘diminishing marginal product of labour’?

Marginal rate of technical substitution

‘Rate you can trade off input for another holding production constant’

\[RTS(K,L) = MP_L/MP_K\]

E.g., for \(q = f(K,L) = L^{1/3}K^{2/3}\)\(MP_L(K,L) =\frac{2}{3} L^{-\frac{2}{3}} K^\frac{2}{3}\) \(MP_K(K,L) =\frac{4}{3} L^{\frac{1}{3}} K{^-\frac{1}{3}}\)

\(MP_L/MP_K = \frac{1}{2} K/L\)

See also ‘elasticity of substitution’

Returns to scale

The rate at which output increases in response to a proportional increase in all inputs.


\[f(\mathbf{tx}) \text{ versus } tf(\mathbf{x})\]

‘Global’ or ‘local’ increasing/decreasing/constant returns to scale; also see ‘elasticity of scale’

Computing… If you know the production function, how do you know if the ‘returns to scale’ are increasing or decreasing?

Slide in a constant \(\alpha>1\) next to each input, simplify, and compare to the original production

E.g.:

\[Q(L,K) = L^{1/4}K^{1/2}\]

\[Q(\alpha L, \alpha K) = (\alpha L)^{1/4}(\alpha K)^{1/2}\] \[=\alpha^{1/4}\alpha^{1/2}L^{1/4}K^{1/2}=\alpha^{3/4}L^{1/4}K^{1/2}=\alpha^{3/4}Q(L,K)\]


\(\rightarrow\) So if we increase inputs by \(\alpha\) here, we increase output by \(\alpha^{3/4}<\alpha\), so DRS everywhere for this production function.

Isoquant maps, rate of technical substitution (RTS)

Cost minimizing input choice

Cost minimizing input choice

Formally a cost function’:

\[c(\mathbf{w},y) \equiv \min_{\mathbf{x}\in R^n_+}(\mathbf{w}\cdot \mathbf{x}) \text{ s.t. } f(\mathbf{x}) \geq y \]

For input price vector \(\mathbf{w>>0}\) and output \(y\).

Lagrangian optimisation w interior solution \(\mathbf{x}^*>>0\) \(\rightarrow\) there is some \(\lambda^*\in R\) s.t.

\(w_i = \lambda \frac{\partial f(\mathbf{x}^*)}{\partial x_i} \forall i\)

\(\rightarrow \frac{\partial f(\mathbf{x}^*)}{\partial x_i}/\frac{\partial f(\mathbf{x}^*)}{\partial x_j} = \frac{w_i}{w_j} \forall i,j\)

I.e., where input mix optimal, each input that is used yields same marginal product per £

Consider a combo of K and L, and the output this yields.


From here, firm can ‘substitute capital for labour’ at rate \(RTS(K,L)= MP_L/MP_K\) (& hold production constant)


If firm uses both K & L and chooses optimally \(\rightarrow\)

Set \(RTS(K,L)\) equal to the input price ratio of these inputs (\(w/r\))


\(\rightarrow\) ‘Same bang for the buck at optimal choices \(K^*\) and \(L^*\)’ i.e.,


\[\frac{MP_K(K^*,L^*)}{r}= \frac{MP_L(K^*,L^*)}{w}\]

Isocost and isoquant

Aside, for intuition and story-telling


If markets for labour and capital are (perfectly) competitive prices of inputs & outputs adjust so that:

  • ‘bang for the buck’ (in revenue) is the same for all inputs and for all production processes

  • Inputs (workers, owners of capital) in every industry will be paid based on their (marginal) productivity …




Advanced question

Back to our ‘replaced by AI’ motivation, a simple model. Suppose:


One product in the economy with ‘constant returns to scale Cobb-Douglas production function:’

\[q = L^{a}K^{1-a}\]


This implies ‘optimising firms will spend a share \(\alpha\) on labour’.


Now suppose technological changes imply \(a\) decreases. What do you think will happen to wage rates, presuming a ‘fixed supply of labour’?

Summing up

Summing up: Optimisation (given a production function and input prices)…

yields a (minimum) cost for every output \(q\) a firm chooses to produce.


\(\rightarrow\) We will be able to construct a cost function

Returns to scale

Returns to scale

Are bigger firms always more efficient? Do things get cheaper to produce the more we produce?

Returns to scale
The rate at which output increases in response to a proportional increase in all inputs.
Constant returns to scale (CRS)
If inputs increase by a factor of X, output increases by a factor equal to X.

E.g., doubling all inputs (labor, capital, land, etc) means exactly doubling all outputs


Increasing returns to scale (IRS)
If inputs increase by a factor of X, output increases by a factor greater than X.


Decreasing returns to scale (DRS)
If inputs increase by a factor of X, output increases by a factor less than X.

Arguments/reasons for scale (dis)economies

IRS

  • Fixed costs (incorporation, buildings, management, planning, R&D) spread over more units

  • Should always be able to at least ‘double everything’ and produce twice as much? (so at least CRS)

  • Scale allows specialisation



Arguments for DRS

  • Limited resources in (relevant) economy; costs begin to rise
  • Managerial issues and coordination problems, bigger ‘centre’ to lobby for favours. See ‘theories of the firm’
  • Harder to give incentives to top manager/CEO?

Costs

Types of costs (‘Basic cost concepts’)

Fixed costs (FC): Costs that must be regularly incurred to remain in business (i.e., for any level of output), but that do not vary with the level of output


Variable costs (VC): Costs that increase with the quantity produced.


Sunk costs: Costs that have been incurred in the past that can never be recovered.

Sunk costs should not enter into any economic decisions.

FC from previous years are sunk costs; FC for future years are not.

Again: Economic profits and cost minimisation; 2-input model

Again: Economic profits and cost minimisation

2-input model

  • Labour costs - wage rates \(w\)
  • Capital costs - rental rate \(v\)

Total costs \(= TC = wL + vK\)

Economic profit = \(\pi\) = Total Revenues - Total Costs

\[\pi = Pq-wL-vK = P\times f(K,L)-wL-vK\]

(P: price of good)

Cost minimizing input choice, expansion path, ratio condition

Choose point where RTS = ratio of input prices

  • RTS = (wage rate/rental rate) = w/v
  • Same ratio of ‘marginal productivity/price’ for each input used

Which point on this curve will minimize cost? . . . The one on the lowest ‘isocost line’ … similar to budget line for consumers

Intuition

\[RTS = MP_L/MP_K = w/v\]

Cost-minimisation for each level of output \(\rightarrow\) firm’s expansion path

  • Total cost curve: from cost of inputs along expansion path

Average and marginal cost curves

Read at home:

  • Average and marginal cost curves

  • … with differing returns to scale: Increasing, decreasing, constant, optimal scale


Skip: “Short and long run” costs for the firm

Average cost
Cost per unit of output

\[AC=TC/q\]

  • TC = ‘summed’ marginal costs and fixed costs
  • AC: ‘average marginal cost’ + FC/q


Marginal cost
Incremental cost of last unit produced
  • I.e., additional cost of producing one more unit of output
  • MC: Slope of (optimising) TC curve at a point

Shape of marginal cost curve depends on production function

  • Constant returns to scale: constant MC (and no FC)

  • Decreasing returns to scale: increasing MC

  • Increasing returns to scale: decreasing MC (and/or constant FC)

On estimating cost/production functions

  • Want to estimate product, firm, and industry production functions

  • Government (regulators etc), forecasters, strategists


Anti-trust, regulating natural monopolies, macro-growth issues and aggregate production, importance of human-K, impact of trade deals (winners/losers), impact of min. wage and labour laws, business strategy and competition, reacting to anticipated market changes…

Production functions \(\Leftrightarrow\) cost functions

  • Amount that can be produced with any set of inputs \(\Leftrightarrow\) (Minimum) cost of producing any output


  • Difficult to estimate these: Lack of publicly available data, lack of exogenous variation in input choices